63 Hoyal Society. 



Integral! with Physical Applications," in the Cambridge Transactions, 

 where a miitaken definition of that term, as used by Green, is given. 



Note IV. 



This theorem may be proved as follows :— 



Let S be any closed surface, containing no part of the electrified bodies 

 within it, which we may conceive to be described between A and B ; let P 

 be the component in the direction of the normal, of the resultant force at 

 any point of the surface S, and let ds be an element of the surface at the 

 same point. Then it may be easily proved (see Math. Joum.vol. iii. p. 204), 

 that JJpds^o, (fl) 



the integrations being extended over the entire surface. Now let S be 

 supposed to consist of three parts ; the portion a, of the surface of A ; its 

 projection /S, on the interior surface of B ; and the surface generated by 

 the curved lines of projection. The value of P at each point of the latter 

 portion of S will be nothing, since the tangent at any point of a line 

 of projection is the direction of the force. Hence, if iJjFds]^ and 

 ( ffFds) denote the values of /yPrfs, for the portions a and ^ of 8, the 

 equation (a) becomes 



l//?ds-] + {//Pds)^0, 



But if p be the intensity of the distribution on the surface A or B, at any 

 point, we have, by Coulomb's theorem, 



P 



Hence 



lffpds]^-{ffpds)^0, 



which is the theorem quoted in the text. 



IX. Proceedings of Learned Societies. 



ROYAL SOCIETY. 



[Continued from vol. vii. p. 626.] 

 May 4, 1854.— Colonel Sabine, R.A.,Treas. and V.P.. in the Chair. 



THE following papers were read : — 

 1. " Account of Researches in Thermo-electricity." By 

 Professor W. Thomson of Glasgow, F.R.S, 



§ I. On the Thermal Effects of Electric Currents in Unequally 



Heated Conductors. 

 Theoretical considerations (communicated in December 1851 to 

 the Royal Society of Edinburgh), founded on observations which 

 had been made regarding the law of thermo-electric force in an un- 

 equally heated circuit of two metals, led me to the conclusion that 

 an electric current must exercise a convective effect on heat in a 

 homogeneous metallic conductor of which different parts are kept at 

 different temperatures. A special application of the reasoning to 

 the case of a compound circuit of copper and iron was made, and it 

 18 repeated here because of the illustration it affords of the mecha- 

 nical principles on which the general reasoning is founded. 



